2 problems

Archer et al.'s conjecture. The number of $\pi\in\overline{\mathcal{Q}}_{\mathcal{M}}$ with $des(\pi)=n$ is equal to

Let $D4AE_n$ denote the symmetric group on $n$ letters. Let $D4AE_n(231)$ be the set of permutations avoiding the pattern $231$, and let $S_n(t)$ be its descent polynomial. Write ……